In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.

is the n × n. It is utilized in the similarity transformation

is an n × n diagonal matrix is called the spectral matrix.

Example

The matrix

has eigenvalues and corresponding eigenvectors

, similar is

One possible choice for an invertible matrix is

are not unique.

Generalized modal matrix

be an n × n matrix. A generalized modal matrix is an n × n matrix whose columns, considered as vectors, form a canonical basis according to the following rules:

• All Jordan chains.

• .

• in order of increasing rank (that is, the generalized eigenvector of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.).

One can show that

is a matrix in Jordan normal form, we obtain

Note that when computing these matrices, equation () is the easiest of the two equations to verify, since it does not require inverting a matrix.

Example

This example illustrates a generalized modal matrix with four Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order. The matrix

with algebraic multiplicity will consist of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see generalized eigenvector.

in Jordan normal form is obtained as follows:

. are not unique.

Notes

References